The timing of a provocation
Casting provokes an attack of opportunity from someone who threatens you when you start casting. Note:
The interrupting event strikes during spellcasting if it comes between when you start and when you complete a spell (for a spell with a casting time of 1 full round or more) or if it comes in response to your casting the spell (such as an attack of opportunity provoked by the spell or a contingent attack, such as a readied action).
There are two options here: either an attack between the start of the spell and its completion, or an attack of opportunity provoked (or readied attack triggered) by the action of casting the spell. That is, the full-round action you take on your turn to begin casting the spell.
So in the case of 1-round-or-longer-casting-time spells, people who move close enough to threaten do not get to take an attack of opportunity just for being there (though obviously they can attack using their turn’s usual attacks). They missed the provoking action that would have allowed them to do so.
Therefore, in this hypothetical, barring something like Spring Attack, he can’t attack while moving, and thus cannot attack the sorcerer. Attacks of opportunity never come into play.
Multiple attacks of opportunity with Combat Reflexes
Moreover, you cannot take multiple attacks of opportunity for a single provoking action, even with Combat Reflexes.
Combat Reflexes and Additional Attacks of Opportunity
[Combat Reflexes] does not let you make more than one attack for a given opportunity, but if the same opponent provokes two attacks of opportunity from you, you could make two separate attacks of opportunity (since each one represents a different opportunity). Moving out of more than one square threatened by the same opponent in the same round doesn’t count as more than one opportunity for that opponent.
Any given action can provoke at most once. This is most relevant when moving about in someone’s threatened area: they get one attack of opportunity, not one for every threatened square you leave for that same action.
No Spring Attack, no threatened area, no attack of opportunity
Without Spring Attack or similar, this is clear:
You threaten all squares into which you can make a melee attack,
Without Spring Attack, the monk cannot attack while moving, and therefore threatens nothing. Since he cannot threaten, he cannot take any attacks of opportunity.
Spring Attack and a threatened area: still need a provocation
Now, the case of Spring Attack (or similar). The monk does threaten, and could take an attack of opportunity if one were provoked. Is one?
Readying an Action
If the triggered action is part of another character’s activities, you interrupt the other character. Assuming he is still capable of doing so, he continues his actions once you complete your readied action.
The charger does not act during the monk’s readied action. As a result, he does not move, and since he does not move, he does not leave any square the monk threatens. Thus, there is no provoking action that the monk can take advantage of.
Provoking the charger
As to whether the charger could have taken an attack of opportunity had the monk not used Tumble to get past him, yes, he could have, because he gets to layer yet another exception here:
Making an Attack of Opportunity
An attack of opportunity “interrupts” the normal flow of actions in the round. If an attack of opportunity is provoked, immediately resolve the attack of opportunity, then continue with the next character’s turn (or complete the current turn, if the attack of opportunity was provoked in the midst of a character’s turn).
So even though the charger does not generally get to act while the monk resolves his readied action (the charger’s charge is put on pause until the monk’s readied action resolves), the charger gets to interrupt the monk’s interrupt, putting his move on pause, and take the attack of opportunity. After the attack of opportunity is resolved, the turn continues, that is, the charger goes back to pause and the monk resolves his move action, and then the charger’s turn continues.
Bonus on attacks of opportunity during a charge
Finally, one last thing to note:
Attacking on a charge
After moving, you may make a single melee attack. You get a +2 bonus on the attack roll and take a −2 penalty to your AC until the start of your next turn.
I would argue that the charger’s attack of opportunity, assuming the monk had not used Tumble, would benefit from this +2. Since the charger has moved, the “after moving” clause is satisfied.
In D&D 3.5, can a character take Attacks of Opportunity while moving without stopping?
In the hypotheticals proposed, no. In general, though? To say “no” conclusively, we would need a rule that explicitly says as much; no such rule exists. To say “yes” conclusively, we would have to demonstrate a situation in which an attack of opportunity happens for a character currently in motion.
Such a situation:
A spellcaster readies an action to cast a spell when someone enters a square within 10 ft. of himself.
An enemy of the spellcaster has the Spring Attack feat, and also has a reach that includes squares 10 ft. from her own position (e.g. Small or Medium with a reach weapon, Large without a reach weapon, etc.). She moves, one way or the other, and her movement takes her into a square 10 ft. from the spellcaster.
The spellcaster’s readied action is triggered, and he takes it. The spring-attacker’s movement is paused so the spellcaster can resolve the readied action.
The spellcaster decides, for whatever reason, not to cast defensively (maybe he does not realize that the spring-attacker has 10-ft. reach, or maybe she has the Mage Slayer feat). Thus, when he casts his spell with his readied action, he is performing an action that provokes.
The spring-attacker, whose movement is on pause while she is in a square 10 ft. from the spellcaster, has a threatened area that includes the spellcaster. She is therefore entitled to take an attack of opportunity, and does so.
The attack of opportunity is resolved; an attack roll is made, and if it is high enough, damage is rolled. If damage is dealt, the spellcaster must make a Concentration check to keep the spell. Regardless, at this point the attack of opportunity has been resolved.
After the attack is complete, the readied action continues. This involves either casting the spell, or simply ending the “pause” if the spellcasting has been disrupted. Either way, the readied action has been resolved.
The spring-attacker’s original movement is unpaused. She now gets to complete her movement.
This is a difficulty that comes up in a lot of RPGs actually, including way back in the early editions of D&D (from which Rogue Trader gets its Disengage-to-avoid-bonus-attack rule).
Think of the round as being full of position and weapon maneuvers and potential attacks, but the roll is only made for the pivotal attack opportunity for which all the the feints and parries and ducks and weaves were softening up the enemy. In this way, the roll is an abstraction of all your careful efforts during the round. See the "Combat Abstractions" sidebar on page 234, where the rules instruct the reader to think of a combat round this way:
[Combatants] are constantly side-stepping, twisting, and ducking, to avoid attacks or assume more favourable combat positions.
And then, when your opponent suddenly flees without carefully disengaging, you can just straight up stab at them because they just gave you a golden opportunity for no effort on your part.
As for the strolling NPC, that's legal in Rogue Trader, apparently as a design choice. The only way you get your "opportunity attack" against a moving target that hasn't used Disengage is if they were already engaged in melee with you, and becoming engaged due to moving only happens when the movement ends adjacent to an opponent (Move p. 241; Engaged in Melee p. 247). Since they're not stopping, they're not engaged, and you don't get the free attack. (Contrast this with another system that uses the "Disengage" design for "opportunity attacks": in early D&D, you become engaged immediately upon becoming adjacent, making opponents "sticky" and enabling them to control their adjacent squares by forcing a Disengage to move again.)
Why that's legal is an aesthetic design choice, as far as I can see. You can rationalise it with a story, or you can object to it on tactical design grounds — but regardless it's RAW. The easiest rationalisation is that opponents who aren't anticipating the need to physically stop someone are unable to do so — they're busy with whatever else they decided to do that round and the opponent ducks and weaves through the battlefield to wherever they're going.
The tactical objection is that this makes it hard to control the battlefield! That's reasonable, but the intuition is likely conditioned by Pathfinder's (unrealistic but functional) way of managing battlefield control.
To accomplish movement control in Rogue Trader, instead of it happening passively as in Pathfinder, you have to use a mix of passive and active tactics. Passively, you can either completely block passage by standing shoulder-to-shoulder, providing no space for an opponent to move through; or you can spread your forces out such that their movement has to end adjacent to one of you. Of course, that passive kind requires a lot of bodies and coordination. Active prevention can be done more solo but with more effort investment, by anticipating the need to stop an advance — which is what the story above implies is the solution — and using the Delay action to attack the incoming opponent. Because attacking someone immediately makes them engaged in melee, they won't be able to continue their movement after your delayed attack, and will furthermore be forced to use Disengage later to continue toward their position objective.
Coming from a D&D point of view...
The game becomes a lot more mobile, speed is more valuable, the concept of front-liner diminishes, and being at range is either difficult or not important.
I tried this for a one-shot once with a couple of new players, and we removed OAs for simplicity's sake. The more experienced players at the table had some fun exploring the consequences of this.
Everyone runs in, attacks, runs away. Both enemies and party members take advantage of cover and we noticed that the disposition of units in the map was much more chaotic.
Speed is more valuable
If mobility is heightened, then speed will improve accordingly. Both the Tabaxi and the Monk appreciated jumping into backlines without hassle and focusing on the squishy ranged goblins that were harassing them.
Frontliners can't frontline as well
One of the reasons a Barbarian or a Fighter stands in front of the party and prevents multiple enemies from going near the Wizards and Sorcerers in the back is their menacing presence and the threat of retaliation. Without AoO, enemies could easily run around and jump into backlines, and the party's melee frontliners just did the same.
Being at a range is more difficult, but at the same time, possibly unnecessary
In closed off areas, where you can't be shooting your bow from 150 ft away, enemies quickly get the jump on ranged attackers, and new tactics are required. If the character is at a range because it does more damage (Dex Fighter with Sharpshooter feat), then they can just step 5 ft from any enemy and attack. If they are at a range because they are very squishy (Wizard with 0 CON modifier), they require new tactics to always be at a distance.
Cat and Mouse
While it didn't happen to us, in open areas, a game of cat and mouse can arise. Goblin dashes to Wizard. Wizard Dashes away. Gobling Dashes to Wizard. Repeat. Because there is no consequence for running away, kiting enemies is easier (you don't take damage for running) in wide spaces.
Overall, it was a fun experience for players, but we decided it didn't work for us. The players enjoyed the fact that, originally, their front-liners were there preventing enemies from running through them. They didn't enjoy enemies now running away and kiting them. But to each their own.
I would say it depends on your expectation. I've only ever played D&D 5e, and I expect our melee combatants to have some ways of preventing enemies from running past them. OAs are one way (a big one, as every character has access to them, from Barbarians to pets). If you don't have such an expectation, I don't see it becoming a big issue. You will simply build your ranged characters expecting to have to handle melee enemies often. All that being said, this is a mere opinion, and to each his own.